Simplify the following expression and state the condition under which the simplification is valid: $a = \dfrac{q^2 - q - 2}{q^2 + q}$
First factor the expressions in the numerator and denominator. $ \dfrac{q^2 - q - 2}{q^2 + q} = \dfrac{(q - 2)(q + 1)}{(q)(q + 1)} $ Notice that the term $(q + 1)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(q + 1)$ gives: $a = \dfrac{q - 2}{q}$ Since we divided by $(q + 1)$, $q \neq -1$. $a = \dfrac{q - 2}{q}; \space q \neq -1$